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PERIODIC MOTION                           325
250.  Effect of Damping on the Period.  Substituting the values of a and 6 in the expression for the period,
+)
(XXVI)
where P0 is the period for the undamped motion. It is evident from equation (XXVI) that the damping increases the period.
VIBRATIONS  ABOUT  A  POSITION   OF  EQUILIBRIUM.
251. Lagrange's Method. In the various pendulum problems which we have discussed the vibrating body was considered to be either a particle or a rigid body. These simplifications were necessary because the methods we have used cannot be applied conveniently to complicated systems. La-grange (1736-1813) introduced into Dynamics a method which can be applied to any vibrating system. The following is a special case of his method adapted to conservative systems which have only one degree of freedom of motion.
Express the potential energy of the system as a function of a properly chosen* coordinate qy so that when expanded in ascending powers of q the first power of q does not appear. Then the potential energy takes the form
U= ft + &22 + ft?3 +    ,         (XXVII)
where ft, ft, etc., are constants.    The constant ft can be
* It is shown in books on advanced Dynamics that such a choice is always possible.